Question

1. Consider the motion of a particle of mass m in a one-dimensional po- tential oO Vo n-CO where ö is the Dirac delta function, and a is the periodic distance of the potential. Prove that COsAa. h2 ka What are k and K?

Answer 1

Given potential V(x) = V_0/a Sigma_n = -infinity^infinity S(x - na) Here S is the Dirac Delta function M rightarrow Mass of particle a rightarrow periodic distance of potential we have the Schrodinger eq^n d^2 Psi/dx^2 + 2M/h^2 (e - V) Psi = 0 = > d^2 Psi/dx^2 + 2ME/h^2 Psi = 2M/h^2 V_0/a Sigma_n = -infinity^infinity S(x - na) Psi Let k = 2ME/h^2, K = 2MV_0/h^2 a (Because integral_- infinity^infinity (x - x_0) dt = 1 = > d^2 Psi/dx^2 + k^2 Psi = k^2 Sigma_n = -infinity^infinity S(x - na) Psi Eigen function Psi_n (x) = 1/squareroot a sin n pi x/2a, n = 2, 4,.... = 1/ squareroot a cos n pix/a n = 1, 3,... From above cos ka + MV_0/h^2 . sin ka/ka = cos Ka